New length operator for loop quantum gravity

TL;DR

This paper introduces a new, background-independent length operator for loop quantum gravity that is well-defined, symmetric, and positive semidefinite, with derivations from geometric and operator substitution perspectives.

Contribution

It presents a novel length operator in loop quantum gravity that is both mathematically consistent and physically meaningful, derived from geometric and operator substitution methods.

Findings

Operator is background independent and well-defined

Derived from geometric averaging and operator substitution

Ensures positive semidefinite and symmetric properties

Abstract

An alternative expression for the length operator in loop quantum gravity is presented. The operator is background independent, symmetric, positive semidefinite, and well defined on the kinematical Hilbert space. The expression for the regularized length operator can moreover be understood both from a simple geometrical perspective as the average of a formula relating the length to area, volume and flux operators, and also consistently as the result of direct substitution of the densitized triad operator with the functional derivative operator into the regularized expression of the length. Both these derivations are discussed, and the origin of an undetermined overall factor in each case is also elucidated.